🤖 AI Summary
To address the challenge of efficiently modeling higher-order interactions in complex systems—particularly protein sequences—this work establishes a rigorous theoretical mapping from restricted Boltzmann machines (RBMs) to the generalized Potts model, yielding the first exact analytical correspondence between the two frameworks. Building on this equivalence, we develop an efficient algorithm grounded in the large-𝑁 statistical approximation, enabling analytical extraction of arbitrary-order effective couplings (e.g., pairwise, three-body). Additionally, we introduce a gauge-fixing formalism to enhance the robustness of parameter estimation. On synthetic data, our method accurately recovers multi-order interactions. When applied to real protein family alignments, it achieves contact map prediction accuracy comparable to state-of-the-art inverse Potts methods. This work thus provides a novel, principled paradigm for modeling higher-order biological networks.
📝 Abstract
Maximum-entropy methods, rooted in the inverse Ising/Potts problem from statistical mechanics, have become indispensable tools for modeling pairwise interactions in disciplines such as bioinformatics, ecology, and neuroscience. Despite their remarkable success, these methods often overlook high-order interactions that may be crucial in complex systems. Conversely, while modern machine learning approaches can capture such interactions, existing interpretable frameworks are computationally expensive, making it impractical to assess the relevance of high-order interactions in real-world scenarios. Restricted Boltzmann Machines (RBMs) offer a computationally efficient alternative by encoding statistical correlations via hidden nodes in a bipartite neural network. Here, we present a method that maps RBMs exactly onto generalized Potts models with interactions of arbitrary high order. This approach leverages large-$N$ approximations, facilitated by the simple architecture of the RBM, to enable the efficient extraction of effective many-body couplings with minimal computational cost. This mapping also enables the development of a general formal framework for the extraction of effective higher-order interactions in arbitrarily complex probabilistic models. Additionally, we introduce a robust formalism for gauge fixing within the generalized Potts model. We validate our method by accurately recovering two- and three-body interactions from synthetic datasets. Additionally, applying our framework to protein sequence data demonstrates its effectiveness in reconstructing protein contact maps, achieving performance comparable to state-of-the-art inverse Potts models. These results position RBMs as a powerful and efficient tool for investigating high-order interactions in complex systems.